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If \(\mathrm{m} = \frac{9}{25}\), \(\mathrm{w} = \frac{15}{32}\), and \(\mathrm{m} + \mathrm{w} + \mathrm{c} = 1\), which of the following gives the values of m, w, and c in increasing order?
Let's break down what we know and what we need to find:
We have three numbers: m, w, and c
Our goal is to arrange these three numbers from smallest to largest and match that order to one of the answer choices.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
To compare fractions easily, let's convert both given fractions to decimals:
For \(\mathrm{m} = \frac{9}{25}\):
Since \(25 \times 4 = 100\), we can write: \(\frac{9}{25} = \frac{9 \times 4}{25 \times 4} = \frac{36}{100} = 0.36\)
For \(\mathrm{w} = \frac{15}{32}\):
Let's divide: \(15 \div 32 = 0.46875\)
We can round this to 0.47 for easy comparison, but let's keep 0.46875 for accuracy.
So we have:
Now we can find c using the equation \(\mathrm{m} + \mathrm{w} + \mathrm{c} = 1\):
\(\mathrm{c} = 1 - \mathrm{m} - \mathrm{w}\)
\(\mathrm{c} = 1 - 0.36 - 0.46875\)
\(\mathrm{c} = 1 - 0.82875\)
\(\mathrm{c} = 0.17125\)
Let's double-check this makes sense: \(0.36 + 0.46875 + 0.17125 = 1.00000\) ✓
So our three values are:
Now let's arrange our decimal values from smallest to largest:
Therefore, in increasing order: c, m, w
Looking at our answer choices, this matches choice A: c, m, w
The values in increasing order are c, m, w.
Answer: A
Verification: \(\mathrm{c} = 0.17125 < \mathrm{m} = 0.36 < \mathrm{w} = 0.46875\), and \(\mathrm{c} + \mathrm{m} + \mathrm{w} = 0.17125 + 0.36 + 0.46875 = 1\) ✓
Faltering Point 1: Misinterpreting "increasing order"
Students may confuse "increasing order" with "decreasing order" and arrange the values from largest to smallest instead of smallest to largest. This fundamental misunderstanding would lead them to select answer choice E (w, m, c) instead of the correct answer A (c, m, w).
Faltering Point 2: Attempting to compare fractions without converting to common form
Students might try to directly compare \(\frac{9}{25}\) and \(\frac{15}{32}\) without converting them to decimals or finding a common denominator, leading to incorrect ordering of m and w from the start.
Faltering Point 1: Decimal conversion errors
When converting \(\frac{15}{32}\) to decimal form, students may make division errors and get an incorrect value (such as 0.48 instead of 0.46875), which could affect the final ordering.
Faltering Point 2: Arithmetic mistakes when calculating c
Students may make computational errors when subtracting \(\mathrm{m} + \mathrm{w}\) from 1, such as: \(\mathrm{c} = 1 - 0.36 - 0.46875 = 0.18125\) instead of 0.17125, potentially changing the relative position of c in the ordering.
Faltering Point 3: Rounding errors affecting comparison
Students might round their decimal values too early in the process (like using 0.47 for w instead of 0.46875), which could lead to incorrect comparisons between closely valued numbers.
Faltering Point 1: Matching the correct ordering to wrong answer choice
Even after correctly determining that the increasing order is c, m, w, students might hastily scan the answer choices and accidentally select a different option that looks similar, such as choice C (m, w, c) or choice B (c, w, m).